The invention relates to a noise damping device for smoothing out and reducing pressure vibrations which occur in automobile fuel feed pumps. It has been proposed heretofore to arrange, for noise damping purposes, devices having an elastic volume within the housing of a fuel feed pump wherein this pump temporarily changes its inside volume, in case of pressure fluctuations, for damping purposes. The formation of pressure vibrations and thus the noise production when conveying a fluid such as fuel that is fed to the internal combustion engine with the aid of a fluid pump is due to the fact that this pump is designed as a piston pump or a pump of related type, e.g., a displacement pump. Such pumps take in the medium to be conveyed, compress it, and discharge it again, with this procedure being repeated cyclically in extremely quick succession. In a roller cell pump as the displacement pump, frequently utilized in fuel feeding, the intake, compression, and discharge procedure of the medium is repeated as often as mutually sealed-off chamber volumes are available, i.e., in case of a five-cell roller cell pump five times. Each of these pumping steps is connected with a short-term drop in the suction vacuum and a rise in the conveying pressure, accordingly the frequency of the thus-produced pressure vibrations is determined from the pump speed of revolution. Therefore, in various vehicles, if the fluid pumps to be damped are fuel feed pumps, noise can also penetrate into the passenger compartment and can be, in part, troublesome, particularly since these noises are due to the pressure fluctuations in the fuel feed system caused by the aforedescribed pump system, which noises are propagated in the form of air-transmitted sound stemming from sound conducted through the material from which the pressure lines are made as well as throughout the automobile car body. A reduction of such pressure vibrations at the site of origin is hardly possible. Consequently, the present invention is directed toward reducing and/or entirely eliminating any disturbing noise by the specific arrangement and construction of damping installations in the fuel circulation system.